Rigging an umbrella-like object
Not sure if this helps or not but here is what I could come up with using only constraints  to simulate an umbrella like opening mechanism.

If you actually need an armature (for exporting purposes, linking, or otherwise) you could probably construct a similar setup with bone constraints on an armature object.



Make a three rod system with a $Post$ (blue), the $Arm$ (white) and the "muscle" or $Piston$ (red) object.



After that add two empties, one at the base will be the $Master$ controlling object, and should lie somewhere along the blue post, the other $Pivot$ will act as a hinge point and should lie somewhere along the arm length.



Then it is all based on a system of parents and constraints. 
The red piston should be parented to the $Master$ controller empty, and constrained with a Track To to the Pivot empty along the $Arm$.



The white $Arm$ should have a Track To constraint to the $Pivot$ empty; 



And lastly the $Pivot$ empty should have a Limit Distance constraint set to the $Master$.



The distance value should exactly match the total length of the "Piston" object (red). You can optionally constrain angles and movements of different parts, like say, constrain the master to move only in $Z$ axis no more than the height of the blue $Post$ rod.



Lastly duplicate the whole system around, making sure you leave the $Post$ and $Master$ out of selection so they remain as single objects



You can then probably use drivers to link the positioning of the $Master$ control object to the scale of any other object, thus achieving what you need.